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<h2 class="hd hd-2 unit-title">W13-PSP1: Power Radiated from Oscillating Charge</h2>
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Power Radiated from Oscillating Charge - part a
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<p><b class="bfseries">(Part a)</b> Consider a positive charge [mathjaxinline]q[/mathjaxinline] that is oscillating sinusoidally, whose position is [mathjaxinline]\vec{r}(t)=y_{0}\cos (\omega t)\hat{j}[/mathjaxinline]&#8212;the motion is nonrelativistic. Find an expression for the magnitude of the electric field [mathjaxinline]|\vec{E}_{\mathrm{rad}}(\vec{r},t)|[/mathjaxinline], at a position [mathjaxinline]\vec{r}[/mathjaxinline], which is at an angle [mathjaxinline]\theta[/mathjaxinline] relative to the [mathjaxinline]\hat{j}[/mathjaxinline] direction. </p>
<p>
Express your answer in terms of <code>k</code> (where [mathjaxinline]k=1/(4\pi \epsilon _0)[/mathjaxinline]), <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>y_0</code> for [mathjaxinline]y_{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], and <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline]. Use <code>abs()</code> for absolute value if needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{E}_{\mathrm{rad}}(\vec{r},t)| =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Power radiated from oscillating charge - part b
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<p><b class="bfseries">(Part b)</b> What is the magnitude of the Poynting vector [mathjaxinline]|\vec{S}(\vec{r},t)|[/mathjaxinline], at a position [mathjaxinline]\vec{r}[/mathjaxinline] and time [mathjaxinline]t[/mathjaxinline]? </p>
<p>
Express your answer in terms of <code>k</code> (where [mathjaxinline]k=1/(4\pi \epsilon _0)[/mathjaxinline]), <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>y_0</code> for [mathjaxinline]y_{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>mu_0</code> for [mathjaxinline]\mu _{0}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\vec{S}_{\mathrm{rad}}(\vec{r},t)| =[/mathjaxinline] </p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Power radiated from oscillating charge - part c
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<p><b class="bfseries">(Part c)</b> Calculate the time-averaged power radiated by this system, which is equal to the time-averaged Poynting flux through a closed surface. Note, the Poynting vector is in the [mathjaxinline]\hat{r}[/mathjaxinline] direction and can be expressed as [mathjaxinline]\vec{S}(\vec{r},t) = |\vec{S}(\vec{r},t)|\hat{r}[/mathjaxinline]. </p>
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Express your answer in terms of <code>k</code> (where [mathjaxinline]k=1/(4\pi \epsilon _0)[/mathjaxinline]), <code>q</code>, <code>c</code>, <code>y_0</code> for [mathjaxinline]y_{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], and <code>mu_0</code> for [mathjaxinline]\mu _{0}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]P =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">W13-PSP2: Transmission/Reflection of S-Polarized Light</h2>
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Transmission/Reflection of S-Polarized Light - part a
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In lecture, we derived the reflection coefficients for p-polarized light. In this problem, you will do the same for s-polarized light. Consider a plane electromagnetic wave that is polarized in the direction perpendicular to the plane of incidence (the electric field only has a component parallel to the interface, shown below for incident light only). </p>
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<p><b class="bfseries">(Part a)</b> Use Maxwell's equations to determine which of the following are the correct relations between the parallel and perpendicular components of the magnetic field? </p>
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<text>a) [mathjaxinline]\dfrac {1}{\mu _1}B_{\perp }^{(1)} = \dfrac {1}{\mu _2}B_{\perp }^{(2)}[/mathjaxinline]</text>
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<text>b) [mathjaxinline]B_{||}^{(1)} = B_{||}^{(2)}[/mathjaxinline]</text>
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<text>c) [mathjaxinline]B_{\perp }^{(1)} = B_{\perp }^{(2)}[/mathjaxinline]</text>
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<text>d) [mathjaxinline]\dfrac {1}{\mu _1}B_{||}^{(1)} = \dfrac {1}{\mu _2}B_{||}^{(2)}[/mathjaxinline]</text>
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Transmission/Reflection of S-Polarized Light - part b
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<p><b class="bfseries">(Part b)</b> Use the definitions above to determine the fresnel reflection and transmission coefficients, [mathjaxinline]r=E_{0R}/E_{0I}[/mathjaxinline] and [mathjaxinline]t=E_{0T}/E_{0I}[/mathjaxinline]. Express your answers in terms of <code>alpha</code> for [mathjaxinline]\alpha[/mathjaxinline] and <code>beta</code> for [mathjaxinline]\beta[/mathjaxinline], which have the familiar definitions below (assuming [mathjaxinline]\mu _{1}=\mu _{2}[/mathjaxinline], which is typically the case): </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\alpha =\dfrac {\cos {\theta _{T}}}{\cos {\theta _{I}}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<td class="equation" style="width:80%; border:none">[mathjax]\beta =\dfrac {n_{2}}{n_{1}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
Hint: use the relation [mathjaxinline]\vec{B}=\left(\dfrac {1}{v}\right)\hat{k}\times \vec{E}[/mathjaxinline], where [mathjaxinline]v=c/n[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]r =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]t =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Transmission/Reflection of S-Polarized Light - part c
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<p><b class="bfseries">(Part c)</b> For mechanical waves, we saw that the reflection and transmission coefficients were related by the equation [mathjaxinline]1 + r = t[/mathjaxinline]. Does this hold for the fresnel reflection and transmission coefficients for s-polarized light? </p>
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